Studies on the Painleve equations

(1)我们成功地用Painleve方程生成了特殊的多项式，推广了第二类Painleve方程的Yablonskii-Vorob‘ev多项式和第四类Painleve方程的Okamoto多项式，特别感兴趣的多项式出现在第三、第五和第六方程的代数解中。(2)对于与第六方程代数解相关的多项式，我们提出了一个猜想，即多项式的每个系数都等于由显式Young图给出的GL的表示次数。(3)上述猜想最近成立。

(1) We succeeded in generating special polynomials by Painleve equations, which generalize the Yablonskii-Vorob'ev polynomial for the second Painleve equation and the Okamoto polynomial for the fourth Painleve equation, Polynomials of particular interest appeared in algebraic solutions of the third, the fifth, and the sixth equations.(2) As for the polynomials associated with algebraic solutions of the sixth equation, we formulated a conjecture that each coefficient of the polynomials is equal to the degree of the representation of GL given by an explicit Young diagram.(3) The above conjecture was recently settled.